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A Robust Whitening Procedure in Blind Source Separation Context Adel BELOUCHRANI and Andrzej CICHOCKI Electrical Engineering Department Ecole Nationale Polytechnique B. P. 182, 16200 El-Harrach, Algiers, Algeria. E-mail: belouchrani@hotmail.com Laboratory for Advanced Brain Signal Processing Brain Science Institute, Riken 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan. E-mail: cia@brain.riken.go.jp Electronics Letters, 2000, Vol.36, pp.2050-2051 Abstract The main objective of this letter is to present an e cient algorithm for robust whitening in the presence of temporally uncorrelated additive noise that may be spatially correlated. This whitening is introduced as a pre-processing step in the blind source separation process. The robust whitening consists in the eigenvalue decomposition of a positive de nite linear combination of a set a correlation matrices taken at nonzero lags. The coe cients of the linear combination are computed by a nite step global convergence algorithm. Some numerical simulations are provided to illustrate the e ectiveness of the solution. Indexing terms: Blind separation of noisy signals, correlation matrices, robust whitening for noisy data. 1 1 Introduction Blind source separation consists of recovering independent signals from their instantaneous mix- tures without any a priori knowledge on these mixtures. Some approaches estimate the source signals by pre-whitening the sensor data followed by a unitary transformation which jointly di- agonalize a set of correlation matrices 1, 2, 3], cumulants matrices 4] or the newly introduced special time frequency distribution matrices 5]. In 1], the whitening matrix is computed from a time-delayed correlation matrix at a delay di erent from zero1 . The authors didn't mentioned details on how this matrix is chosen. In their simulations a correlation matrix at a lag close to the zero lag was chosen to ensure the positive de niteness of the matrix. Note that in this case, we can not neglect the in uence of the noise unless it is perfectly white. Moreover by choosing a time lag close to zero we need a high sampling frequency rate. In this letter, we solve this problem by estimating the whitening matrix from an eigenvalue decomposition of a positive de nite matrix which is linear combination of correlation matrices taken at nonzero lags. The coe cients of the linear combination are determined by using the nite step global convergence algorithm proposed in 6]. 2 Data Model In the context of blind source separation, a n-dimensional observation vector x(t) is assumed to be generated by x(t) = As(t) + n(t) (1) where A 2 IRn n is the unknown nonsingular mixing matrix, s(t) is a n-dimensional vector of source signals that are assumed mutually uncorrelated and temporally correlated. The vector n(t) is an additive noise assumed to be zero mean, temporally white and independent from the source signals. Contrary to classical assumptions, no assumption is made on either its distribution or its spatial correlation properties. Hence, its covariance matrix E n(t)n(t)T ] = Rn can be full matrix which is generally unknown. Under the above assumptions, the correlation 1 The idea of the estimation of the whitening matrix from correlation matrices at delay di erent from zero was rst suggested in 2] (See Section III.3 page 436) and worked out in 3]. 2 matrices of the observation take the following structure: Rx(0) = E x(t)x(t)T ] = ARs(0)AT + Rn (2) Rx(i) = E x(t)x(t ; i)T ] = ARs (i)AT for i = 1 0: (3) The problem under consideration is how to estimated the mixing matrix A up to a permutation matrix and a diagonal matrix using only the observed noisy data x(t). 3 The solution The Second Order Blind Identi cation (SOBI) algorithm, that uses a linear combination of correlation matrices for the estimation of the whitening matrix, is as follows: 1. Estimate the correlation matrices and compute a singular value decomposition of the n K matrix set = Rx(1) R ^ Rx(K )], ^ = UR VT R (4) where UR IRn n and V IRnK nK are orthogonal matrices, and has nonzero entries at (i i) 2 2 position (1 i n) and zeros elsewhere. 2. For i = 1 K compute Fi = UT Rx(i)UR : R^ (5) 3. Choose any initial 2 IRn . 4. Compute K X F= i Fi (6) i=1 Test: Compute a Schur decomposition of F 2 IR n n . If F is positive de nite, go to step 5, otherwise go to Update. Update: Choose an eigenvector u corresponding to the smallest eigenvalue of F and update by replacing with + d, where d = uuTF11uu uuTFK uu]] T T T jj F FK jj (7) and go to step 4. This loop is completed in a nite number steps (See 6] for proof). 5. Compute K X C= i Rx (i) ^ (8) i=1 3 and perform an eigenvalue decomposition (EVD) of C: C = Ucdiag 2 1 n ]UT 2 c (9) where 2 's are the singular values of C. A whitening matrix is given by i W = diag 1 n ];1 UT : c (10) 6. Form the whitened correlation matrices: Rx(i) = WRx(i)WT ^ ^ for i = 1 K: (11) 7. A unitary matrix U is obtained as joint diagonalizer of the set Rx (i) i = 1 ^ f j K g. 8. The source signals are estimated as ^(t) = UT Wx(t) s (12) and the mixing matrix is estimated as A = W;1U: ^ (13) The use of this algorithm for windowed correlation matrices of nonstationary signals as in 1] or for cumulant matrices is straightforward. 4 Simulation results The simulated sources are two Gaussian signals located in the frequency domain around the normalized frequency 0.5 and 0.6, respectively. Two sensors are considered. The p-th element of the q-th column of the mixing matrix A is e2i p q with 1 = 0:2 and 2 = 0 4. The additive noise is generated from a zero mean and temporally white Gaussian process with the following covariance matrix, 2 3 Rn = 2 41 5 1 where 2 is the noise power and is the coe cient of noise correlation. We consider here 4 cor- P relation matrices. Figures 1 and 2 display the rejection level de ned as n(n1;1) p6=q j(A# A)pq j2 ^ versus the Signal to Noise ratio (SNR) (for = 0:9) and versus the coe cient of the noise corre- lation ( ) (for SNR = 10 dB), respectively. These plots shows the performance of SOBI when the whitening uses correlation matrix (full line) and when it uses a combination of a set correlation 4 matrices at time lag di erent from zero. The plots show a signi cant increase in performance (approaching 8 dB) when whitening with the combination of a set of the correlation matrices at time lag di erent from zero. 5 Conclusion The blind source separation method presented in 1] and based on the same idea as SOBI 2, 3], estimates a whitening matrix from a time-delayed correlation matrix at a delay di erent from zero. As this matrix is not guaranteed to be positive de nite for some delays, a whitening matrix can not be always computed from the suggested correlation matrix. To solve this problem, we suggest in this letter to use a positive de nite linear combination of a set of correlation matrices taken at nonzero time lags to compute the whitening matrix. The coe cients of this linear com- bination are computed by a nite step global convergence algorithm. The provided simulations show signi cant increase in performance of BSS algorithms when the robust whitening with the combination of the set of the correlation matrices at time lags di erent from zero is used. References 1] S. Choi and A. Cichocki,\Blind separation of nonstationary sources in noisy mixtures," IEE Elec- tronics Letters, vol. 36, no. 9, pp. 848-849, April 200 2] A. Belouchrani and K. Abed Meraim and J.-F Cardoso and E. Moulines, \A blind source separation technique using second order statistics," IEEE Trans. on SP, vol. 45, pp. 434{444, Feb. 1997. 3] A. Belouchrani ,\Separation autodidacte de sources: Algorithme, Performances et Application a des signaux experimentaux", Doctorate thesis report. ENST 95 E 014. Telecom Paris, July 1995. 4] J.-F. Cardoso and A. Souloumiac, \An e cient technique for blind separation of complex sources," in Proc. IEEE SP Workshop on Higher-Order Stat., Lake Tahoe, USA, 1993. 5] A. Belouchrani and M. G. Amin, \Blind source separation based on time-frequency signal represen- tation," IEEE Trans. on SP, vol. 46, pp. 2888{2898, Nov. 1998. 6] L. Tong, Y. Inouye and R. Lui, \A Finite-step global convergence algorithm for the parameter estimation of multichannel MA processes," IEEE Trans. on SP, vol. 40, no. 10, pp. 2547{2558, Oct. 1992. 5 −5 −− Whitening using a linear combination of correlation matrices − Whitening using the auto correlation matrix −10 −15 Mean rejection level (dB) −20 −25 −30 −35 −40 0 2 4 6 8 10 12 14 16 18 20 SNR (dB) Figure 1: Mean rejection level vs SNR for coe cient of noise correlation of = 0:9. 6 −15 −− Whitening using a linear combination of correlation matrices − Whitening using the autocorrelation matrix −20 Mean rejection level (dB) −25 −30 −35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Coefficient of noise correlation Figure 2: Mean rejection level vs coe cient of noise correlation for SNR = 10 dB. 7

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